A characterization of subsets of curves of finite length and the analyst’s TSP

We characterize subsets of Hilbert space that are contained in a curve
of finite length. We do so by extending and improving results of Peter
Jones and Kate Okikiolu for sets in $\R^d$. Their results formed the
basis of quantitative rectifiability in $\R^d$.

In the talk we will explain the following statement which we obtain: given
a set $K$ , $\diam(K) +\sum\beta^2(Q)diam(Q) \sim \ell(\Gamma_{MST})$.
Here $\beta$ (the Jones $\beta$ number) is taken with respect to $K$, the
sum is over a multiresolutional family of (overlapping) balls $Q$ centered
on $K$, and $\Gamma_{MST}$ is the shortest connected set containing $K$.